Two thin circular discs of mass m and 4m, having radii and , respectively, are rigidly fixed by a massless, rigid rod of length through their centres. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so t

Question

Two thin circular discs of mass m and 4m, having radii $a$ and $2 a$ , respectively, are rigidly fixed by a massless, rigid rod of length $l = \sqrt{24} a$ through their centres. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $ω$ . The angular momentum of the entire assembly about the point 'O' is $\vec{L}$ (see the figure). Which of the following statement(s) is(are) true [2016]

Smart Achievers · Accepted Answer

(1, 3)

Position of CM on the axis of rod.

$x_{C M} = \frac{m (0) + 4 m (l)}{m + 4 m} = \frac{4 l}{5}$

$\cos θ = \frac{1}{\sqrt{l^{2} + a^{2}}} = \frac{\sqrt{24 a}}{\sqrt{a^{2} + 24 a^{2}}} = \frac{\sqrt{24}}{5}$ $[∵ l = \sqrt{24 a} given]$

$O A = \sqrt{{(2 l)}^{2} + {(2 a)}^{2}} = \sqrt{96 a^{2} + 4 a^{2}} = 10 a$

Let complete system rotates about z-axis with a constant angular velocity $ω'$

$∴$ $\frac{ω'}{ω} = \frac{2 π (2 a)}{2 π (10 a)} \Rightarrow ω' = \frac{ω}{5}$

Magnitude of angular momentum of the system about its center of mass

$L_{C M} = I_{C M} ω = [\frac{m a^{2}}{2} + \frac{4 m {(2 a)}^{2}}{2}] ω = \frac{17}{2} m a^{2} ω$

Magnitude of angular momentum of CM of system about point O.

$L' = 5 m \times \frac{9 l ω}{5} \times \frac{9 a}{5} = \frac{81 l m ω a}{5} = \frac{81 \sqrt{24} m ω a^{2}}{5}$

Magnitude of z-component of angular momentum of system about point O

$L_{z} = L' \cos θ - L_{C M} \sin θ$

$= \frac{81 \sqrt{24} m ω a^{2}}{5} \times \frac{\sqrt{24}}{5} - \frac{17}{5} m a^{2} ω \times \frac{1}{5}$

$= m a^{2} ω (\frac{1944}{25} - \frac{17}{10})$

Solution

1 The centre of mass of the assembly rotates about the z-axis with an angular speed of $ω / 5$

2 The magnitude of angular momentum of centre of mass of the assembly about the point O is $81 m a^{2} ω$

3 The magnitude of angular momentum of the assembly about its centre of mass is $\frac{17}{2} m a^{2} ω$

4 The magnitude of the z-component of $\vec{L}$ is $55 m a^{2} ω$

Want Us to Email you About Special Offers & Updates?

Solution

1 The centre of mass of the assembly rotates about the z-axis with an angular speed of ω/5

2 The magnitude of angular momentum of centre of mass of the assembly about the point O is 81ma2ω

3 The magnitude of angular momentum of the assembly about its centre of mass is 172 ma2ω

4 The magnitude of the z-component of L→ is 55 ma2ω

Want Us to Email you About Special Offers & Updates?

1 The centre of mass of the assembly rotates about the z-axis with an angular speed of $ω / 5$

2 The magnitude of angular momentum of centre of mass of the assembly about the point O is $81 m a^{2} ω$

3 The magnitude of angular momentum of the assembly about its centre of mass is $\frac{17}{2} m a^{2} ω$

4 The magnitude of the z-component of $\vec{L}$ is $55 m a^{2} ω$